Properties

Degree 1
Conductor 73
Sign $0.997 - 0.0679i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 − 0.866i)2-s i·3-s + (−0.5 + 0.866i)4-s + (−0.965 + 0.258i)5-s + (−0.866 + 0.5i)6-s + (0.707 + 0.707i)7-s + 8-s − 9-s + (0.707 + 0.707i)10-s + (0.258 + 0.965i)11-s + (0.866 + 0.5i)12-s + (0.965 + 0.258i)13-s + (0.258 − 0.965i)14-s + (0.258 + 0.965i)15-s + (−0.5 − 0.866i)16-s + (−0.707 − 0.707i)17-s + ⋯
L(s,χ)  = 1  + (−0.5 − 0.866i)2-s i·3-s + (−0.5 + 0.866i)4-s + (−0.965 + 0.258i)5-s + (−0.866 + 0.5i)6-s + (0.707 + 0.707i)7-s + 8-s − 9-s + (0.707 + 0.707i)10-s + (0.258 + 0.965i)11-s + (0.866 + 0.5i)12-s + (0.965 + 0.258i)13-s + (0.258 − 0.965i)14-s + (0.258 + 0.965i)15-s + (−0.5 − 0.866i)16-s + (−0.707 − 0.707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.997 - 0.0679i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.997 - 0.0679i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(73\)
\( \varepsilon \)  =  $0.997 - 0.0679i$
motivic weight  =  \(0\)
character  :  $\chi_{73} (7, \cdot )$
Sato-Tate  :  $\mu(24)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 73,\ (1:\ ),\ 0.997 - 0.0679i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.8843917006 + 0.03009699073i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.8843917006 + 0.03009699073i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6957503700 - 0.2381840963i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6957503700 - 0.2381840963i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−31.69256685154389389704665396587, −30.45086287591734763343125481109, −28.53330579912489197729985180981, −27.723095698533816039422593532912, −26.87898724508447856475291774059, −26.30351027462310641068686978963, −24.76723060774371092484892252544, −23.63526178284193893116958135842, −22.92387040097028510610109892253, −21.36346168773775840331000183471, −20.09922080517630425268707084244, −19.15401365465555377406971388219, −17.499957601557947946238997678748, −16.61087366230569736864191821218, −15.651559744988601481299643354515, −14.79080875291029559033658906626, −13.52825616159151943483909076755, −11.2067008429608068841280148387, −10.594039239569802132934473387615, −8.71776901707641266631811455317, −8.255376883203085097743006008186, −6.46768784286789951794947056426, −4.844087069114438208931145620135, −3.86667442188692715377466458267, −0.59809977602160066443202796601, 1.37685065250904311620182589381, 2.791340741001799144218778662024, 4.54107521198267993873202523577, 6.81458518908115740360569877054, 8.05009998183072880654266061038, 8.90918425141482516257286505424, 10.93220068511939668271787605968, 11.78689414606800258768561079091, 12.58535625172870812269525368204, 14.02985826729847243314629617747, 15.50512990550333813240021057124, 17.26383914819916339926534793700, 18.24041891849468578695667472558, 18.98657216808285461861727288379, 19.99544044427905338541366130760, 21.06229215375930774392829492923, 22.65347881210654549470603934666, 23.40391443690179833487235313247, 24.89169222559200574116662392622, 25.8755285633908682541706699581, 27.31292648540897826794280111593, 28.065165089388778103094854650195, 29.09413846394984232305146953640, 30.41947710244288138803444804817, 30.91479526243548049257036202883

Graph of the $Z$-function along the critical line